Bibliotheca Polyglotta

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. If A:B=C:D, then mA:nB=mC:nD.

四幾何。其第一與二。偕第三與四。比例等。第一、第三、同任為若干倍。第二、第四、同任為若干倍。則第一所倍、與第二所倍。第三所倍、與第四所倍。比例亦等。

For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D; I say that, as E is to G, so is F to H.

For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.

Since E is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken, therefore K is the same multiple of A that L is of C. [V. 3] For the same reason M is the same multiple of B that N is of D. And, since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N, if it is equal, equal, and if less, less. [V. Def. 5] And K, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H; therefore, as E is to G, so is F to H. [V. Def. 5]

Therefore etc. Q. E. D.

本卷界　\\　說六
一系。凡四幾何。第一與二。偕第三與四。比例等。卽可反推第二與一。偕第四與三。比例亦等。何者。如上倍甲之壬、與倍乙之子。偕倍丙之癸、與倍丁之丑。等、大、小、俱同類。而顯甲與乙、若丙與丁。卽可反說。倍乙之子、與倍甲之壬。偕倍丁之丑、與倍丙之癸。等、大、小、俱同類。而乙與甲。亦若丁與丙。本卷界　\\　說六
二系。別有一論。亦本書中所恆用也。曰。若甲與乙、偕丙與丁。比例等。則甲之或二或三倍、與乙之或二、或三倍。偕丙之或二、或三倍、與丁之或二、或三倍。比例俱等。倣此以至無窮。

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